Approximation to $\int_{-1}^{1}f(x)dx$ using only $f(0)$, $f'(-1)$, and $f''(1)$

Question

The exercise is

Using only $f(0)$, $f'(-1)$, and $f''(1)$, compute an approximation to $\int_{-1}^{1}f(x)dx$ that is exact for all quadratic polynomials

I have only seen guassian quadrature without derivatives so i'm not sure how to go about this. Usually I check what conditions are needed so my approxiation is exact for all monomials of degree $\leq 4$, but since it asks to use derviatives of the function i'm not sure what to do.'

Could anyone lend a hand?

Answer 1

HINT If $f(x) = ax^2 +bx+c$, you have $$ \int_{-1}^1 f(x) dx = \frac{2a}{3} + 2c, $$ can you compute $f(0)$ and $f''(1)$ and complete the problem?